The family of mixed Normal distributions has been used extensively in econometric and statistical theory. Members of this class share the property of being Normal conditionally on a random variable. The latter is called the mixing variate because its density acts as a weight function in mixing the conditional Normals into the unconditional (marginal) density. The most famous mixed Normal is Student's t which is obtained by using a Gamma-distributed mixing variate. Here, we will consider another type of mixing variate: functionals of Brownian motions. Though they are commonly referred to, the marginal densities that arise are yet to be derived and plotted. Still unexplored are features like quantiles, fmiteness of the density throughout its support, and existence of its moments. Such features also have implications for the validity, design, and interpretation of Monte Carlo (MC) studies.
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